International Journal on Recent and Innovation Trends in Computing and Communication
Volume: 4 Issue: 5
ISSN: 2321-8169
292 - 293
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A Study on Primitive Ring
Tulasi Prasad Nepal
Central Department of Mathematics,
Tribhuvan University
Kirtipur, Kathmandu, Nepal
Email: [email protected] or [email protected]
Abstract: In this paper we discus about prime ring, simple ring, primitive ring and some lemmas and theorems.
Key words: simple ring, Prime ring, Reduced Primitive ring, division ring.
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1. INTRODUCTION
Throughout the paper, the word 'ring' means an associative
ring (but not necessarily commutative ring with an identity
element 1). The word 'subring' always means a subring
containing the identity element of the larger ring. If R= {0},
R is called the zero ring, it is noted that this is the case if and
only if 1= 0 in R. If R ≠ {0} and ab=0 implies either a= 0 or
b= 0, R is said to be domain and if R is commutative then R
is called an integral domain. Without exception the word
'ideal' refers to a 2- sided ideal. First we give an example of
an element x in a ring R such that Rx  xR. Let R be the
ring of 2×2 upper triangular matrices over a nonzero ring k
1 0
𝑘 0
and let
x=
. Then
Rx=
and xR=
0 0
0 0
𝑘 𝑘
, therefore Rx  xR.
0 0
A ring R is called prime if aRb= (0) implies either a= 0 or
b= 0. So a ring R is a prime if 0 is prime ideal of R.
𝑍 𝑛𝑍
Example of a prime ring, for integer n ≠ 0, R=
𝑍 𝑍
is a prime ring. It is a subring of
S= M2(Z). Note that
nS  R. If a, b ∈ R such that aRb= 0. Since S is a prime
ring. We conclude that a=0 or b= 0, we know that a ring R is
prime if and only if Mn(ℝ) is prime, where Mn(ℝ) is the set
of n × n matrices with entries from ℝ.
A nonzero ring R is said to be a simple if (0) and R are the
only ideal in R. A ring R is said to be a reduced ring if R has
no nonzero nilpotent elements or equivalently if a 2= 0
implies a= 0. For instance, the direct product of any family
of domain is reduced. An ideal generated by a single
element is called a principal ideal. For example the trivial
ideal 0 and the ideal R are both principal because 0= (0) and
R= (1).
2. DEFINITION
An ideal P of a ring R is called left primitive if it is the
largest ideal of R contained in some maximal left ideal M of
R. Thus P= [R: M]= { r ∈ R : rR  M}. This definition is
not symmetrical, and it is known that left primitive does not
imply right primitive. None the less; the attribute left is
usually omitted. A ring is called primitive if 0 is a primitive
ideal. It easily follows that, for any ideal P,
𝑅
𝑃
is a primitive
ring if and only if P is primitive ideal.
Let R be a ring. A left R- module is an additive abelian
group M together with an action (called scalar
multiplication) 𝜇: R×M→ M(the image 𝜇(r, x) being
denoted by rx) such that for all r, s ∈ R and x, y ∈ M:
(i)
r (x+ y)= rx+ ry
(ii)
(r+ s) x= rx+ sx
(iii)
r (sx)= (rs) x
It is denoted by RM.
Let M be
an R- module . A non empty subset N of M is called an
R- submodule of M if and only if
(i)
x, y ∈ N implies x- y ∈ N
(ii)
x ∈ N and r ∈ R implies rx ∈ N.
Let R, k be two rings and M = RVk be an (R, k) – bimodule . We write E= End (Vk), which operates on V from
the left. We say that R acts densely on V k if, for any f ∈ E
and any v1, v2, . . ., vn ∈ V, there exists r ∈ R such that rvi= f
(vi) for i= 1, 2, . . . ,n. A module M (≠{0}) is said to be
simple if and only if the only submodule of M are {0} and
M.
A left R- module RA is called faithful if whenever rA= 0
implies r= 0, r ∈ R. In other words A is faithful if for any 0
≠ r, r ∈ R, rA ≠ 0.
3. SOME LEMMAS AND THEOREMS
3.1 Lemma: A ring R with identity is a domain if and only if
R is prime and reduced.
Proof: Assume a ring R is a domain. Then an= 0 implies a=
0, so R is reduced. Also if aRb=0 implies ab= 0 which
implies a= 0 or b= 0. So R is a prime.
Conversely, assume R is prime and reduced. Let
a, b ∈ R
such that ab= 0. Then for any r ∈ R,
(bra)2=
brabra= br(ab) ra= 0, so bra= 0. This means that bRa= 0, so
b= 0 or a= 0 since R is prime.
3.2 Lemma: Every simple ring is a primitive ring.
292
IJRITCC | May 2016, Available @ http://www.ijritcc.org
_______________________________________________________________________________________________
International Journal on Recent and Innovation Trends in Computing and Communication
Volume: 4 Issue: 5
ISSN: 2321-8169
292 - 293
_______________________________________________________________________________________________
Proof: Let R be a simple ring. Then 0 is a maximal ideal of
R. Let M be a maximal left ideal of R Then 0 is the largest
ideal contained in M. Hence 0 is a primitive ideal of R
which shows that R is primitive ring.
3.3 Lemma: Every maximal ideal is a primitive ideal.
Proof: Let I be a maximal ideal of R. Then
ring which implies that
𝑅
𝐼
𝑅
𝐼
is a simple
is a primitive ring since every
simple ring is primitive which implies that I is a primitive
ideal of R.
3.4 Lemma: A commutative ring is primitive if and only if it
is a field.
Proof: Given R is a commutative ring. Assume R is a
primitive ring. Then 0 is a primitive ideal of R which
implies that 0 is the largest ideal of R contained in some
maximal left ideal M of R since R is a commutative. So we
have 0= [R: M], where
[R: M]= { a ∈ R : aR 
M}, since P is a primitive ideal of R if and only if P= [R
:M]
Now let a ∈ M. Then aR  M implies a ∈ [R: M]. But [R:
M]= 0, so a= 0.
Therefore 0=M implies 0 is a maximal ideal of R. Since R is
commutative ring. So R is a field.
Conversely, Assume R is a field. We know that a field has
no proper ideals. So 0 is a primitive ideal of R which implies
that R is a primitive ring.
3.4 Theorem: Let P be a proper ideal of a commutative ring
R. Then P is a prime if and only if whenever aRb  P
implies a ∈ P or b ∈ P, a, b ∈ R.
Proof: Since P is a proper ideal of R. Assume that P is a
prime and aRb  P (a, b ∈ R). Then
(RaR) (RbR)  P, since P is a two sided ideal & RaR is a
principal ideal of R generating by a ∈ P. This implies RaR
 P or RbR  P, since P is prime. So 1.a.1 ∈ P or 1.b.1
∈ P which implies a ∈ P or b ∈ P
Conversely, Let A and B be two ideals of R such that AB
 P. If possible A  P, then there exists a ∈ A such that a
 P. But aRb  AB  P for all b ∈ B, which implies
aRb  P for all b ∈ B.
Since a  P so we have b ∈ P implies B  P. So P is a
prime ideal of R
.
3.5 Lemma: A primitive ring is prime.
Proof: Let R be a primitive ring. So 0 is a primitive ideal.
Let aRb= 0. There exists a faithful simple left R- module A.
Then RbA is a submodule of A. Hence RbA= 0 or RbA= A,
since A is simple. If RbA= 0 then bA= 0 if R= 1 which
implies b= 0, since A is faithful. If RbA= A, then aRbA= aA
but aRb= 0. Therefore aA= 0 implies a= 0 since A is
faithful. This implies that 0 is a prime ideal of R, so R is a
prime ring.
3.6 Corollary: A primitive ideal of a ring R is prime ideal.
Proof: Let P be a primitive ideal of a ring R. Then
primitive ring which implies that
𝑅
𝑃
𝑅
𝑃
is a
is prime ring. This
implies P is a prime ideal of R.
3.7. [2] Density Theorem: Let R be a ring and V be a semisimple left R module. Then for
k= End (RV),
R acts densely on Vk.
3.8. Theorem: Let R be a left primitive ring such that
a
(ab- ba)= (ab- ba) a for all a, b ∈ R. Then R is a division
ring.
Proof: Let RV be a faithful simple R- module with k=
End (RV). It sufficies to show that
dim RV= 1 (For
then R ≅ k which is a division ring). Assume instead
there exist k- linearly independent vectors u, v ∈ V. By
density theorem there exist a, b ∈ R such that au= u,
av= 0 and bu= 0, bv= u. But then a (ab- ba) (v)= a2u= u,
(ab- ba) a (v)= 0, a contradiction.
3.9. Theorem: Let R be a left primitive ring such that 1+ a2
is a unit for any a ∈ R. Then R is a division ring.
Proof: We repeat the argument. If the independent
vectors u, v ∈ V exist, we can find a ∈ R such that a
(u)= -v and a (v)= u. Then (1+ a2) (u)= u- u= 0 which
contracts the assumption.
REFERENCES:
[1] Goodearl K. R., Ring Theory (Non singular rings and
Modules), Marcel Dekker, Inc. New York, 1976
[2] Lam, T.Y., A First Course in Noncommutative Rings,
Springer- Verlag, 1991
[3] Jacobson, N. “Structure of Rings,” Amer. Math. Soc.
Colloquium Publications Vol. 37, Providence, R.I., 1964
[4] Small L. W. Review in Ring Theory(as priented in Math
Reviews, (1940- 79, 1980- 84) Amer. Math. Soc.
Providence, R. I. 1981, 1986.
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